Random matrices: tail bounds for gaps between eigenvalues

Hoi Nguyen, Van Vu, and myself have just uploaded to the arXiv our paper “Random matrices: tail bounds for gaps between eigenvalues“. This is a followup paper to my recent paper with Van in which we showed that random matrices of Wigner type (such as the adjacency matrix of an Erdös-Renyi graph) asymptotically almost surely had simple spectrum. In the current paper, we push the method further to show that the eigenvalues are not only distinct, but are (with high probability) separated from each other by some negative power of . This follows the now standard technique of replacing any appearance of discrete Littlewood-Offord theory (a key ingredient in our previous paper) with its continuous analogue (inverse theorems for small ball probability). For general Wigner-type matrices (in which the matrix entries are not normalised to have mean zero), we can use the inverse Littlewood-Offord theorem of Nguyen and Vu to obtain (under mild conditions on ) a result of the form

for any and , if is sufficiently large depending on (in a linear fashion), and is sufficiently large depending on . The point here is that can be made arbitrarily large, and also that no continuity or smoothness hypothesis is made on the distribution of the entries. (In the continuous case, one can use the machinery of Wegner estimates to obtain results of this type, as was done in a paper of Erdös, Schlein, and Yau.)

for (one also has good results of this type for smaller values of ). This is only optimal in the regime ; we expect to establish some eigenvalue repulsion, improving the RHS to for real matrices and for complex matrices, but this appears to be a more difficult task (possibly requiring some quadratic inverse Littlewood-Offord theory, rather than just linear inverse Littlewood-Offord theory). However, we can get some repulsion if one works with larger gaps, getting a result roughly of the form

for any fixed and some absolute constant (which we can asymptotically make to be for large , though it ought to be as large as ), by using a higher-dimensional version of the Rudelson-Vershynin inverse Littlewood-Offord theorem.

In the case of Erdös-Renyi graphs, we don’t have mean zero and the Rudelson-Vershynin Littlewood-Offord theorem isn’t quite applicable, but by working carefully through the approach based on the Nguyen-Vu theorem we can almost recover (1), except for a loss of on the RHS.

As a sample applications of the eigenvalue separation results, we can now obtain some information about eigenvectors; for instance, we can show that the components of the eigenvectors all have magnitude at least for some with high probability. (Eigenvectors become much more stable, and able to be studied in isolation, once their associated eigenvalue is well separated from the other eigenvalues; see this previous blog post for more discussion.)

Is it possible to extend the results to some class of random (infinite dimensional) hermitian operators (e.g. to show that the probability of a spectrum gap less than is bounded by whenever is bounded by a fixed positive power of a sufficiently small )?

I think one could make some artificial infinite-dimensional random operator model for which the currently known techniques would extend, but for the models that people care the most about (e.g. discrete or continuous random Schrodinger operators), these sorts of methods aren’t yet applicable (they rely too much on having all of the entries having non-zero variance, and so don’t cope well with the band-limited or otherwise localised nature of operators such as Schrodinger operators). But there is certainly an effort to push a lot of the Wigner theory in the direction of more physically relevant matrices and operators, such as discrete random Schrodinger operators (e.g. there are now some results on band-limited matrices which are sort of an interpolant between the random Schrodinger models and the Wigner models).

I wonder if you could comment about analogous statements for non-Hermitian matrices? For example, suppose you generate a random graph as in G(n,p) but now throwing in directed edges. Would the results you describe here (simple spectrum, no eigenvector with zero entry) still hold and how difficult would they be to prove?

The techniques in this paper break down (among other things, they rely on the Cauchy interlacing law, which breaks down in non-Hermitian settings), but I have a student looking into this problem who is trying some other methods.

So, if we look at specific types of matrices then randomly and asymptomatically they will likely fall into a small spectrum of distinct eigenvalues, potentially with the sign flipped. If this is symmetric you can look for those eigenvalues and find those types of matrices? Where might that be useful? Or is that not within the scope of study?

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